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Properties

Label 2-336-112.101-c2-0-42
Degree $2$
Conductor $336$
Sign $0.132 - 0.991i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.19 + 1.60i)2-s + (1.67 − 0.448i)3-s + (−1.14 + 3.83i)4-s + (9.00 + 2.41i)5-s + (2.71 + 2.14i)6-s + (5.79 + 3.92i)7-s + (−7.51 + 2.73i)8-s + (2.59 − 1.50i)9-s + (6.88 + 17.3i)10-s + (0.273 − 0.0733i)11-s + (−0.200 + 6.92i)12-s + (−16.0 − 16.0i)13-s + (0.636 + 13.9i)14-s + 16.1·15-s + (−13.3 − 8.78i)16-s + (−4.94 − 2.85i)17-s + ⋯
L(s)  = 1  + (0.597 + 0.802i)2-s + (0.557 − 0.149i)3-s + (−0.286 + 0.958i)4-s + (1.80 + 0.482i)5-s + (0.452 + 0.358i)6-s + (0.828 + 0.560i)7-s + (−0.939 + 0.342i)8-s + (0.288 − 0.166i)9-s + (0.688 + 1.73i)10-s + (0.0248 − 0.00666i)11-s + (−0.0167 + 0.577i)12-s + (−1.23 − 1.23i)13-s + (0.0454 + 0.998i)14-s + 1.07·15-s + (−0.835 − 0.549i)16-s + (−0.290 − 0.167i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.132 - 0.991i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.132 - 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.61329 + 2.28703i\)
\(L(\frac12)\) \(\approx\) \(2.61329 + 2.28703i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 1.60i)T \)
3 \( 1 + (-1.67 + 0.448i)T \)
7 \( 1 + (-5.79 - 3.92i)T \)
good5 \( 1 + (-9.00 - 2.41i)T + (21.6 + 12.5i)T^{2} \)
11 \( 1 + (-0.273 + 0.0733i)T + (104. - 60.5i)T^{2} \)
13 \( 1 + (16.0 + 16.0i)T + 169iT^{2} \)
17 \( 1 + (4.94 + 2.85i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-7.51 + 28.0i)T + (-312. - 180.5i)T^{2} \)
23 \( 1 + (1.42 - 0.821i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (24.6 - 24.6i)T - 841iT^{2} \)
31 \( 1 + (33.4 + 19.2i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (17.8 - 66.4i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 4.84T + 1.68e3T^{2} \)
43 \( 1 + (36.5 + 36.5i)T + 1.84e3iT^{2} \)
47 \( 1 + (-23.0 + 13.2i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-56.8 + 15.2i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-2.98 - 11.1i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (5.01 - 18.7i)T + (-3.22e3 - 1.86e3i)T^{2} \)
67 \( 1 + (-8.72 - 32.5i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 + 57.4iT - 5.04e3T^{2} \)
73 \( 1 + (-24.5 + 42.5i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-23.7 - 41.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (87.0 + 87.0i)T + 6.88e3iT^{2} \)
89 \( 1 + (-40.2 - 69.7i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 76.8iT - 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.75463284128021151181517224542, −10.48109766502221798095907659427, −9.422656805545298075737525186319, −8.755132234724343828474010381591, −7.51731508240652753452190379850, −6.74131216945096759230926275738, −5.45317402888830440075825862091, −5.06291160831958362901659428721, −3.00448768378905960205417551758, −2.17530047473271745364695481251, 1.63993323574345436687085653230, 2.18757326525069858777469301636, 3.98647679702104778149653851355, 4.99710512900919660170570905859, 5.84010820168915601146844611176, 7.18053560422263064028095106628, 8.736145312042998370555271899620, 9.579247192869802828465741077796, 10.05572764174967268837656348842, 11.03934226562120047395317990960

Graph of the $Z$-function along the critical line